Pressure gradients, fluid contacts and capillary equilibrium

Quick refresher

In a continuous phase at rest, pressure grows linearly with depth, with a gradient $\rho g$ . Where the oil and water lines cross, the pressures of the phases are equal and the capillary pressure (the pressure difference between oil and water that comes from the small pores) is zero. This is the free water level. The crossing of the gas and oil lines is the gas–oil contact (GOC).
In a water-wet rock (one that the water prefers to coat), the formation tester shows the pressure of the oil (or gas) wherever this phase is continuous. That is the oil-bearing part, and the transition zone (the interval above the OWC where the water saturation falls with height) down to the OWC. Below the OWC, including the interval between the OWC and the FWL, it shows the pressure of water.
The OWC from logs lies above the FWL by $h_d = p_d/(\Delta\rho\,g)$ . This is from a fraction of a metre in good sandstones to metres and tens of metres in tight rocks.
The error of a contact is the uncertainty of the two lines at the depth where they cross, divided by the difference of the gradients. It grows with the scatter of the points, with the distance of extrapolation (how far a line is extended beyond its points), and with a smaller density difference of the phases.
Errors common to all the points of one survey do not shift the crossing relative to that well. A constant error of the gauge does not change it, and a common shift of depths moves it together with the logs of the same well. If the lines are taken from different wells, the absolute accuracy of the gauges and of the depths enters the error: metres instead of fractions of a metre.
Logs see hydrocarbons, but the pressure shows the water line: the hydrocarbons are immobile, or the rock is oil-wet (the oil clings to the grains instead of the water). Shifted parallel lines are separate systems with their own contacts. Lines that coincide are needed for the layers to be connected during development, but they do not prove it.

Key relationships

z^* = z_0 + \dfrac{p_o(z_0) - p_w(z_0)}{G_w - G_o}

the crossing of the oil and water lines (the FWL) from their pressures at a reference depth $z_0$

\sigma_{z^*} = \dfrac{\sqrt{\sigma_o^2(z^*) + \sigma_w^2(z^*)}}{G_w - G_o}

standard error of the contact; $\sigma^2(z) = s^2\left[\tfrac1n + \tfrac{(z - \bar z)^2}{\sum (z_i - \bar z)^2}\right]$ for each line

h_d = \dfrac{p_d}{(\rho_w - \rho_o)\,g}

height of the OWC above the FWL; $p_d$ is the entry pressure at reservoir conditions

\rho_{w,\text{res}} = \rho_{w,\text{sc}}/B_w

density of formation water in the reservoir, from its density in the analysis and its formation volume factor (the ratio of its volume in the reservoir to its volume at the surface)

Common pitfalls

Drawing one line through points of different systems, or through supercharged and unstabilised measurements.
Extending a short line for tens of metres without allowing for how its uncertainty grows with the distance from the mean depth of the points.
Taking the crossing of the lines for the OWC from logs, or requiring them to coincide.
Taking different OWCs from logs in two wells as proof that the blocks are isolated. The cause may be different rocks or a paleo-contact (an old contact, left from an earlier position of the oil–water boundary).
Comparing pressures from different wells without allowing for the absolute accuracy of the gauges and depths, and for the time of the measurements.

After this lesson you can

explain which phase's pressure the formation tester records in the oil-bearing part, in the transition zone and below the OWC, and why the crossing of the lines gives the free water level and not the OWC from logs
find the FWL and the gas–oil contact from lines built by least squares, and turn the errors of pressures, gradients and depths into the error of the contact
judge whether there are enough points: their number, their spread in depth, the distance of extrapolation and the density contrast of the phases
compare the FWL with the OWC from logs through the entry pressure of the rock, and recognise from the difference immobile (residual) hydrocarbons and a missed pay zone (a layer that can produce)
build a water line from regional data (the density of water from a laboratory analysis and the pressure in a neighbouring well) and allow for the errors that this brings
recognise separate fluid systems and hydraulically isolated compartments, and understand what a common pressure line does and does not prove

Which pressure the formation tester sees at a contact

In an accumulation that is not being produced, the fluids have come to equilibrium over geological time. Each continuous phase (gas, oil, water) follows the hydrostatic law for its own density (the fluid is at rest). Its pressure grows with depth along a straight line with the slope $\rho g$ (fluid gradients). The oil and water lines cross at the free water level: there the pressures of the phases are equal and the capillary pressure is zero. Above the FWL, the oil pressure exceeds the water pressure by $(\rho_w - \rho_o)\,g\,h$ , where $h$ is the height above the FWL. This excess is the capillary pressure (saturation profile).

Oil enters the rock only when the capillary pressure exceeds the entry pressure $p_d$ of the largest connected channels. So from the FWL up to a height $h_d = p_d/(\Delta\rho\,g)$ the rock is fully saturated with water. The top of this interval is the OWC that the resistivity log (a log that shows oil by its high electrical resistance) sees. Above it lies the transition zone, where the oil is already continuous and the water saturation decreases with height.

The formation tester cannot choose a phase: it measures the pressure of the liquid at the probe. The experience of many surveys in wells drilled with water-based mud is this. In a water-wet rock, the tool shows the oil pressure in the oil-bearing part and in the whole transition zone, and the water pressure below the OWC. There are two explanations. First, at the wall the filtrate has pushed the oil out to the residual saturation (the small amount of oil that cannot be pushed out). The capillary pressure in the flushed zone has almost vanished. The pressure of the liquid in it equals the pressure of the continuous oil beyond the invasion front (the edge of the zone that the filtrate has entered). Second, the probe with the line of the tool is a huge “pore” compared with the channels of the rock. In equilibrium, the large pores of a water-wet rock are filled with the non-wetting phase (the phase that does not coat the grains), which is the oil, and they are at its pressure. Between the OWC and the FWL there is no oil in the rock, and the tool shows the water pressure there, although the oil line, extended downwards, lies still farther to the right.

The pressure that the formation tester records at the oil–water contact

p_{\text{meas}}(z) = \begin{cases} p_o(z) = p_{\text{FWL}} + G_o\,(z - z^*), & z < z_{\text{OWC}} \\ p_w(z) = p_{\text{FWL}} + G_w\,(z - z^*), & z > z_{\text{OWC}} \end{cases}

where:

$p_{\text{meas}}(z)$ stabilised pretest pressure at depth $z$ , Pa
$z$ true vertical depth, measured downwards, m
$z^*,\ p_{\text{FWL}}$ depth of the free water level and the pressure at it (the pressures of the phases are equal), m, Pa
$G_o,\ G_w$ pressure gradients of oil and water, $\rho_o g$ and $\rho_w g$ , Pa/m
$z_{\text{OWC}}$ oil–water contact from logs, $h_d = p_d/(G_w - G_o)$ above the FWL, m

Units: SI (in practice MPa, kPa/m and m)

Holds when

The accumulation is in hydrostatic and capillary equilibrium, and the formation water does not flow.
The rock is water-wet, and the oil in the transition zone is continuous. In an oil-wet rock the picture is the mirror image (advanced section).
The measurements are valid ones, without supercharging and without an unfinished build-up (classes of pretests). The well is drilled with water-based mud.

At the level of the OWC, the recorded pressure drops abruptly by $p_d$ : above it the tool sees oil, below it water. The extensions of both lines cross below the OWC, at the FWL. In a good reservoir rock the jump is a few kilopascals, and it is hard to tell from the scatter of the points. In a tight rock it is larger, but supercharging gets in the way there (Supercharging and permeability from formation-tester pretests).

Small differences are easy to see on an excess pressure plot. From each measurement we subtract the pressure on the water line at the same depth. Water then falls on the vertical line of zero. Oil falls on an inclined straight line that comes to zero at the FWL. Offsets of a few kilopascals, which are invisible on a “pressure–depth” plot in megapascals, become visible. The same is done with any reference line: with the oil gradient, or with a calculated gradient of formation water.

CalculatedThe lower part of the oil rim in the new well (the lesson's model). Left: water saturation from the model capillary pressure curve, with an entry pressure of 7 kPa. Right: excess pressure over the water line. The dots are pretests, the solid line is the pressure that the tool records, and the dashed line is the oil line extended to the FWL.

The contact from the crossing of lines

The basics (two points per zone and the crossing of straight lines) are in the topic on reservoir pressure. Building a line by least squares from a set of pretests is in the topic on the formation tester. Here is the continuation: how to find the crossing of two fitted lines, and how far we can trust it. The order of work is this. Reject the measurements that are not valid. Divide the points into zones (by the kind of fluid that the gradients, logs and samples suggest). Fit a line in each zone. Only then look for the crossings.

Depth of the crossing of two pressure lines

z^* = z_0 + \frac{p_a(z_0) - p_b(z_0)}{G_b - G_a}

where:

$z^*$ depth of the crossing: the FWL for an oil–water pair, the GOC for a gas–oil pair, the GWC for a gas–water pair, m
$z_0$ a reference depth that is convenient for the calculation, usually near the expected contact, m
$p_a(z_0),\ p_b(z_0)$ pressures on the lines of the light ( $a$ ) and heavy ( $b$ ) phases at the reference depth, Pa
$G_a,\ G_b$ gradients of the light and heavy phases, Pa/m

Units: any consistent system (MPa, MPa/m and m, or kPa, kPa/m and m)

Holds when

Both lines belong to one hydraulically connected system in equilibrium.
The densities of the phases are constant with height. If the composition changes noticeably with depth, the line of the oil or gas is curved (section on fluid systems).

The pressure difference of the two lines changes with depth at the rate $G_b - G_a$ . So the crossing lies at a distance of “pressure difference divided by gradient difference” from the reference depth. The smaller the density contrast, the farther each kilopascal of error shifts the contact.

Gradient contrast and the sensitivity of the contact to a pressure error

Pair of fluids: water – light oil
Densities, g/cm³: 1.07 and 0.72
Gradient difference, kPa/m: 3.43
Shift of the contact per 1 kPa, m: 0.29

Pair of fluids: water – heavy oil
Densities, g/cm³: 1.07 and 0.90
Gradient difference, kPa/m: 1.67
Shift of the contact per 1 kPa, m: 0.60

Pair of fluids: water – gas
Densities, g/cm³: 1.07 and 0.20
Gradient difference, kPa/m: 8.53
Shift of the contact per 1 kPa, m: 0.12

Pair of fluids: oil – gas
Densities, g/cm³: 0.72 and 0.20
Gradient difference, kPa/m: 5.10
Shift of the contact per 1 kPa, m: 0.20

Pair of fluids: volatile oil – gas condensate
Densities, g/cm³: 0.55 and 0.40
Gradient difference, kPa/m: 1.47
Shift of the contact per 1 kPa, m: 0.68

Pair of fluids	Densities, g/cm³	Gradient difference, kPa/m	Shift of the contact per 1 kPa, m
water – light oil	1.07 and 0.72	3.43	0.29
water – heavy oil	1.07 and 0.90	1.67	0.60
water – gas	1.07 and 0.20	8.53	0.12
oil – gas	0.72 and 0.20	5.10	0.20
volatile oil – gas condensate	0.55 and 0.40	1.47	0.68

The densities are approximate, at reservoir conditions. The gradient is $9.81\rho$ kPa/m. A volatile oil is a very light oil that gives off a lot of gas, and a gas condensate is a gas that drops a light liquid when its pressure falls. With the same accuracy of measurements, the GOC of light oil and gas is found more accurately than the FWL. The contact of volatile oil with gas condensate, and the FWL of heavy oil, are found noticeably worse.

Pretests of the new well at the contact and in the water-bearing part

Depth, m: 2535.5
Pressure, MPa: 27.9227
Mobility, mD/(mPa·s): 6.5
Where: transition zone, control point

Depth, m: 2539.5
Pressure, MPa: 27.9474
Mobility, mD/(mPa·s): 38
Where: below the OWC, above the FWL, control point

Depth, m: 2544.0
Pressure, MPa: 27.9957
Mobility, mD/(mPa·s): 46
Where: water-bearing part

Depth, m: 2550.5
Pressure, MPa: 28.0624
Mobility, mD/(mPa·s): 130
Where: water-bearing part

Depth, m: 2556.0
Pressure, MPa: 28.1221
Mobility, mD/(mPa·s): 28
Where: water-bearing part

Depth, m: 2562.5
Pressure, MPa: 28.1890
Mobility, mD/(mPa·s): 75
Where: water-bearing part

Depth, m: 2569.0
Pressure, MPa: 28.2565
Mobility, mD/(mPa·s): 11
Where: water-bearing part

Depth, m: 2575.5
Pressure, MPa: 28.3264
Mobility, mD/(mPa·s): 52
Where: water-bearing part

Depth, m	Pressure, MPa	Mobility, mD/(mPa·s)	Where
2535.5	27.9227	6.5	transition zone, control point
2539.5	27.9474	38	below the OWC, above the FWL, control point
2544.0	27.9957	46	water-bearing part
2550.5	28.0624	130	water-bearing part
2556.0	28.1221	28	water-bearing part
2562.5	28.1890	75	water-bearing part
2569.0	28.2565	11	water-bearing part
2575.5	28.3264	52	water-bearing part

A continuation of the survey from the topic Pressure measurement with the wireline formation tester: the same well, with water-based mud. The pressures are absolute, the depths are vertical, and all the measurements are valid. The two control points (points kept only as a check) are not included in the lines. The data are assumed for illustration.

Worked exampleThe free water level in the new well, and its error

Given (assumed data)

Oil line: nine valid pretests from the topic Pressure measurement with the wireline formation tester: $n = 9$ , $\bar z_o = 2501.06$ m, $\sum(z - \bar z)^2 = 3092$ m², $G_o = 7.065$ kPa/m, $s_o = 1.02$ kPa
Water line: six pretests of the water-bearing part from the table above
Reference depth: $z_0 = 2540$ m

Solution

The water line by least squares: $n = 6$ , $\bar z_w = 2559.58$ m, $\sum(z - \bar z)^2 = 688.7$ m², $G_w = 10.490$ kPa/m, $s_w = 0.98$ kPa. The density of water is $10.490/9.81 = 1.069$ g/cm³.
Pressures on the lines at 2540 m: oil 27.95406 MPa, water 27.95326 MPa. The oil is higher by 0.80 kPa.
The difference of the gradients is $10.490 - 7.065 = 3.425$ kPa/m. The crossing is $z^* = 2540 + 0.80/3.425 = 2540.2$ m.
The uncertainty of the lines at this depth: oil $1.02\sqrt{1/9 + (2540.2 - 2501.06)^2/3092} = 0.79$ kPa, water $0.98\sqrt{1/6 + (2540.2 - 2559.58)^2/688.7} = 0.83$ kPa.
The standard error of the FWL: $\sqrt{0.79^2 + 0.83^2}/3.425 = 0.33$ m. An approximate 95% interval is ±0.7 m.
The control points. At 2535.5 m the pressure is above the water line by 16.6 kPa and lies on the oil line: this is the transition zone, where the oil is continuous. At 2539.5 m the pressure is below the oil line by 3.1 kPa and differs from the water line by 0.6 kPa: there is only water there.

Answer: The FWL is 2540.2 ± 0.33 m (one standard error). Within the error it agrees with the estimate from two points per zone in the topic on reservoir pressure (about 2540.3 m).

What it means. The accuracy of a third of a metre comes from many points in each zone, a small scatter and a short extrapolation of the water line. The contribution of the oil line is almost as large, although it has more points: it is extended by 39 m from the mean depth of its points. A constant error of the gauge does not affect the crossing, because both lines shift in the same way. A common shift of the depths of one survey moves the FWL together with the logs of this well, and it does not change the comparison with the OWC. The control points confirm the picture: the tool sees oil down to the OWC and water below it. The data in this example are assumed for illustration.

Contact error, and how many points we need

Each fitted line is known best at the mean depth of its points, and the worse the farther we go from it. The uncertainty of the pressure on a line at depth $z$ is made up of two parts: the error of the mean level, and the error of the slope multiplied by the distance from the mean depth. The errors of the two lines are independent. So their variances (the squares of the standard errors) at the depth of the crossing add up, and we convert the result into metres by dividing by the difference of the gradients.

Standard error of a contact from two lines

\sigma_{z^*} = \frac{\sqrt{\sigma_a^2(z^*) + \sigma_b^2(z^*)}}{|G_b - G_a|},\qquad \sigma^2(z) = s^2\left[\frac1n + \frac{(z - \bar z)^2}{\sum_i (z_i - \bar z)^2}\right]

where:

$\sigma_{z^*}$ standard error of the depth of the crossing, m
$\sigma_a(z^*),\ \sigma_b(z^*)$ standard errors of the pressure on each line at the depth of the crossing, Pa
$s,\ n,\ \bar z$ scatter of the points about the line, number of points and their mean depth (each line has its own), Pa, —, m
$z_i$ depths of the points of the line, m

Units: any consistent system (in practice kPa, kPa/m and m)

Holds when

The errors of the points are random, independent and of equal size. The points within each line belong to one phase.
Systematic errors are not included. A constant error of the gauge in one survey shifts both lines in the same way and does not change the crossing. A common shift of depths moves the crossing together with the logs of the same well. For lines from different wells, these errors are added separately (section on the regional water line).
With few points, $s$ is estimated only roughly. The confidence interval (the range in which the true value is expected to lie) is taken with Student's coefficient for $n - 2$ degrees of freedom, and not just ±2σ (about the standard error).

The error of the contact grows with the scatter of the points, and it falls as $1/\sqrt n$ with their number. It depends strongly on how far the line has to be extended. The second fraction in the bracket grows quickly when the crossing lies outside the interval of the points.

CalculatedThe crossing of the oil and water lines of the new well near the FWL, on a plot of excess over the fitted water line. The bands are ±2 standard errors of each line. The bar on the axis is ±2 standard errors of the FWL from the example.

How many points do we need? The fluid density from a gradient is found more accurately the more points we have and the wider the interval between the extreme ones. But with few points the main trouble is different: the scatter $s$ is estimated from the points themselves. With one or two degrees of freedom (the number of points minus the number of fitted parameters) the estimate is unreliable, and Student's coefficient is huge. The table shows the half-width of the 95% interval of the density for evenly spaced points with a scatter of 1 kPa.

Half-width of the 95% interval of fluid density from a gradient, g/cm³ (scatter of the points 1 kPa)

Number of points: 3
Interval 10 m: 0.183
Interval 20 m: 0.092
Interval 40 m: 0.046

Number of points: 5
Interval 10 m: 0.041
Interval 20 m: 0.021
Interval 40 m: 0.010

Number of points: 10
Interval 10 m: 0.023
Interval 20 m: 0.012
Interval 40 m: 0.006

Number of points	Interval 10 m	Interval 20 m	Interval 40 m
3	0.183	0.092	0.046
5	0.041	0.021	0.010
10	0.023	0.012	0.006

The points are evenly spaced; Student's coefficient is for $n - 2$ degrees of freedom. Three points give an interval about 4.5 times wider than five points on the same depth interval.

How to space the pretests for a contact

1
At least four or five valid points per line
Otherwise the scatter cannot be estimated, and any error of one measurement goes into the slope. We add repeat measurements at neighbouring depths, because some of the points will turn out not to be valid.
2
The whole thickness of the zone
The extreme points weigh more than the middle ones: doubling the interval halves the error of the gradient.
3
Closer to the contact on both sides
A short extrapolation is more important than extra points far from the contact. But in the transition zone and in tight layers near the contact, the risk of supercharging is higher.
4
A thin zone
If there are not enough points for a line, we take the fluid density from a sample and draw a line with the known gradient through the points that we have. This is the forced gradient technique.

The free water level and the OWC from logs

A comparison of the level from pressure with the contact from logs is one of the main checks of a survey. In an equilibrium water-wet accumulation, the OWC lies above the FWL by $h_d$ . In high-permeability sandstones this is tens of centimetres, and in tight rocks metres and tens of metres (height of the OWC above the FWL). A difference of this size and this sign is normal. Any other difference needs an explanation.

Worked exampleDo the FWL, the OWC from logs and the core agree

Given (assumed data)

FWL from pressure: from the previous example, 2540.2 ± 0.33 m
OWC from the resistivity log: 2538.2 ± 0.2 m (the water saturation reaches one)
Mercury injection (a laboratory test that forces mercury into the rock) on samples of the basal sandstone: entry pressure 0.125 MPa; mercury: 480 mN/m, contact angle 140°
Reservoir oil–water pair: interfacial tension (the tension of the surface between oil and water) 24 mN/m, contact angle 30°; the gradient difference is 3.425 kPa/m

Solution

The distance from the OWC to the FWL: $2540.2 - 2538.2 = 2.0$ m, with an error $\sqrt{0.33^2 + 0.2^2} = 0.39$ m. The OWC is reliably above the FWL.
The entry pressure that follows from the pressure measurements and the log: $p_d = 3.425\cdot 2.0 = 6.85$ kPa (±1.3 kPa).
The entry pressure from core, converted to reservoir fluids (conversion of the curve): the wetting tensions are $24\cos 30^\circ = 20.78$ and $480\,|\cos 140^\circ| = 367.7$ mN/m. Then $p_d = 125\cdot 20.78/367.7 = 7.06$ kPa.
The height of the OWC above the FWL from core: $h_d = 7.06/3.425 = 2.1$ m, against 2.0 m from pressure and the log.

Answer: The entry pressure from pressure and the log is 6.85 ± 1.3 kPa, and from core it is 7.06 kPa. Three independent sources agree.

What it means. The difference of two metres between the OWC and the FWL is explained by capillary forces, and there is no need to look for other causes. The calculation of hydrocarbons in place follows the saturation profile above the FWL, not a flat surface at the OWC. If the OWC from logs had come out clearly below the FWL, that would point to immobile oil or to a measurement error (below). The data in this example are assumed for illustration.

Why the OWC from logs and the FWL from pressure differ

Cause: Capillary forces; rock of different quality
OWC relative to the FWL: above by $h_d$ ; in tight layers by metres
How to check: capillary pressure curves from core; a common FWL for layers of different quality

Cause: Immobile hydrocarbons below a paleo-contact
OWC relative to the FWL: below: the log sees oil, the pressure shows the water line
How to check: a sample or a test of the interval gives only water

Cause: Oil-wet rock in the transition zone
OWC relative to the FWL: below: the transition zone is below the FWL
How to check: wettability from core; the oil here is mobile

Cause: A separate hydrodynamic system
OWC relative to the FWL: any
How to check: shifted pressure lines, faults and sealing layers on the section

Cause: Flow of water in the aquifer (the water-bearing formation)
OWC relative to the FWL: the contact is tilted and differs between wells
How to check: levels in different wells, the water gradient (Perched and tilted contacts, trapped water)

Cause	OWC relative to the FWL	How to check
Capillary forces; rock of different quality	above by $h_d$ ; in tight layers by metres	capillary pressure curves from core; a common FWL for layers of different quality
Immobile hydrocarbons below a paleo-contact	below: the log sees oil, the pressure shows the water line	a sample or a test of the interval gives only water
Oil-wet rock in the transition zone	below: the transition zone is below the FWL	wettability from core; the oil here is mobile
A separate hydrodynamic system	any	shifted pressure lines, faults and sealing layers on the section
Flow of water in the aquifer (the water-bearing formation)	the contact is tilted and differs between wells	levels in different wells, the water gradient (Perched and tilted contacts, trapped water)

Errors in the measurements also give a difference. Supercharging of tight water-bearing layers shifts the water line to the right and the FWL upwards. An error in matching the depths of the log and of the formation tester shifts the levels relative to each other.

Immobile oil below the contact. An accumulation could have formed with the contact in a different position. Afterwards the structure tilted, or a block dropped along a fault, or part of the oil escaped, or formation water rose. The lower part of the former accumulation went through imbibition (the soaking up of water by the rock), and the oil was left there as disconnected droplets at the residual oil saturation (imbibition profile). The resistivity log sees hydrocarbons in this interval, while the pressure shows the water line: the oil is not continuous and does not transmit its own pressure. The lower boundary of such a zone is the paleo-contact, the trace of the former OWC. The oil in it is not recovered by ordinary development and does not count as moveable hydrocarbons (those that can flow to the wells). So the interval between the FWL and the contact from logs is checked with a formation tester sample (Reservoir fluid sampling) or with a test.

The opposite case: a missed pay zone. Sometimes the pressure shows an oil line where the resistivity log does not see oil. This happens in fine-grained shaly and silty sandstones with a high residual water saturation (a lot of water that stays in the pores). It also happens in sections with fresh formation water, where the resistivities of oil-bearing and water-bearing rock are close (Water saturation in shaly sands). An oil gradient in such an interval is a strong argument for mobile oil, and we check it with a sample.

A tar mat. At the base of some oil accumulations lies a layer where the oil has turned into bitumen (a very heavy, tar-like oil) that hardly moves. This layer is called a tar mat. Usually the asphaltenes (the heaviest, tar-like components of oil) that dropped out of the oil have collected there. On a survey it looks like a series of “dry” pretests with no inflow between the oil-bearing and the water-bearing parts. Such a layer can separate the oil from the edge water (the aquifer outside the oil accumulation), and this matters for estimating water influx during development (Aquifer models: water influx).

A water line from regional data

A well may not penetrate the water-bearing part, or it may penetrate it in tight rocks, where pretests do not stabilise or are supercharged. Then we build the water line from outside data: the density of formation water from analysis of samples, and the pressure in the aquifer from measurements in a neighbouring well that penetrated the same aquifer. The density of water measured at standard conditions is converted to reservoir conditions through the water formation volume factor. The mass is the same, and the volume in the formation is larger by a factor of $B_w$ (at usual reservoir conditions, heating expands water more than pressure compresses it).

Density and gradient of formation water from analysis

\rho_{w,\text{res}} = \frac{\rho_{w,\text{sc}}}{B_w},\qquad G_w = \rho_{w,\text{res}}\,g

where:

$\rho_{w,\text{sc}}$ density of a water sample at standard (laboratory) conditions; from salinity — Formation water and formation temperature, kg/m³
$B_w$ formation volume factor of water: the volume in the formation per unit volume at standard conditions; for waters without gas it is usually 1.00–1.05, m³/m³
$G_w$ pressure gradient of water in the formation, Pa/m
$g$ acceleration of gravity, 9.81, m/s²

Units: SI; in metric form $G_w$ [kPa/m] $= 9.81\,\rho_{w,\text{res}}$ [g/cm³]

Holds when

Dissolved gas is ignored (otherwise the density is calculated from the mass of water and gas).
Salinity and temperature are the same over the interval through which the line is extended.

An error in density of 0.005 g/cm³ is 0.05 kPa/m. Over a 20 m extension of the line it gives about 1 kPa. This is not the danger. The danger is a shift of the level of the line, which the pressure and depth from another well bring.

Worked exampleFWL in a well on the flank from the regional water line

Given (assumed data)

Well 2, pretests of the oil-bearing part (depth, m — pressure, MPa): 2488.0 — 27.5857; 2494.5 — 27.6335; 2501.0 — 27.6781; 2508.5 — 27.7326; 2515.0 — 27.7762; 2522.0 — 27.8270; all valid. The water-bearing part here is tight, its pretests are suspected of supercharging, and they are not used for the water line
Water from a laboratory analysis: 1.092 g/cm³ at 20 °C, $B_w = 1.020$ ; density error ±0.005 g/cm³
Pressure in the water-bearing part: in the first well: 28.163 MPa at 2560.0 m (topic Reservoir pressure and temperature)
Accuracy in each well: gauge ±7 kPa, true vertical depth ±1.0 m (one standard error)

Solution

The oil line of well 2: $G_o = 7.068$ kPa/m, $s = 1.07$ kPa; at 2540 m it is 27.95406 MPa.
The water line: $\rho_w = 1.092/1.020 = 1.0706$ g/cm³, $G_w = 9.81\cdot 1.0706 = 10.503$ kPa/m; at 2540 m $28.163 - 0.010503\cdot 20 = 27.95294$ MPa.
The crossing: the oil is above the water by 1.12 kPa, the difference of gradients is 3.435 kPa/m, and the FWL is $2540 + 1.12/3.435 = 2540.3$ m.
The error budget of the pressure difference at the FWL (the separate errors that add up): the oil line at this depth, 1.4 kPa; the water density, $9.81\cdot 0.005 = 0.049$ kPa/m over 19.7 m of extension, 1.0 kPa; two gauges, $7\sqrt2 = 9.9$ kPa; the depths of two wells, $\sqrt{(7.068\cdot 1)^2 + (10.503\cdot 1)^2} = 12.7$ kPa.
The total is $\sqrt{1.4^2 + 1.0^2 + 9.9^2 + 12.7^2} = 16.2$ kPa, and the standard error of the FWL is $16.2/3.435 = 4.7$ m.

Answer: The FWL in well 2 is 2540.3 ± 5 m. Within the error it agrees with the FWL of the new well, but it is known 14 times more roughly.

What it means. Within one well, the errors of the gauge and of the depths are common to both lines and cancel. With lines from different wells they remain and give almost the whole error. The scatter of points and the water density are secondary here. We can reduce the error by getting the water level in well 2 itself, from valid water-bearing pretests or from supercharged ones with the supercharging allowed for (Supercharging and permeability from formation-tester pretests). We can also improve the depth matching, using the directional survey and the elevations of the rotary table (the rig floor). There is a separate danger in time. If a neighbouring field is depleting the aquifer, the pressure in it falls between the surveys. A line built from an old measurement must then first be brought to the date of the new survey. The data in this example are assumed for illustration.

Several fluid systems and isolated blocks

An accumulation can consist of several systems that are not connected to each other. These are formations separated by continuous shale seals, or blocks separated by impermeable faults (breaks in the rock along which the layers have moved; sealing faults). Each such system is a hydraulically isolated compartment with its own pressure lines and its own contacts. On a “pressure–depth” plot it shows as parallel lines of one fluid shifted against each other, or as pressure steps at the boundaries of formations. A shift larger than the error of comparison is a sign of separate systems.

The reverse statement is not true. Coinciding lines show only that the pressures have equalised over geological time. A fault or a thin layer with a very low conductance has time to equalise pressure over millions of years, but it hardly lets fluid through over the years of development. Connection during development is checked by tests: interference tests (a change at one well, and watching the pressure at another), and watching the pressure in neighbouring blocks (Partially communicating faults, Pressure measurements in producing reservoirs: barriers and accuracy).

Pretests of well 3 behind the fault

Depth, m: 2488.5
Pressure, MPa: 27.4905
Where: oil-bearing part

Depth, m: 2493.0
Pressure, MPa: 27.5210
Where: oil-bearing part

Depth, m: 2498.5
Pressure, MPa: 27.5617
Where: oil-bearing part

Depth, m: 2503.0
Pressure, MPa: 27.5923
Where: oil-bearing part

Depth, m: 2507.5
Pressure, MPa: 27.6237
Where: oil-bearing part

Depth, m: 2516.0
Pressure, MPa: 27.7015
Where: water-bearing part

Depth, m: 2523.5
Pressure, MPa: 27.7791
Where: water-bearing part

Depth, m	Pressure, MPa	Where
2488.5	27.4905	oil-bearing part
2493.0	27.5210	oil-bearing part
2498.5	27.5617	oil-bearing part
2503.0	27.5923	oil-bearing part
2507.5	27.6237	oil-bearing part
2516.0	27.7015	water-bearing part
2523.5	27.7791	water-bearing part

All the measurements are valid; the pressures are absolute, and the depths are vertical. The data are assumed for illustration.

Worked exampleA block behind a fault: the same oil, another system

Given (assumed data)

Well 3: five pretests in the oil-bearing part and two in the water-bearing part (the table above)
Reference lines of the field: oil and water of the new well (the first example): $G_o = 7.065$ , $G_w = 10.490$ kPa/m
Accuracy of comparing different wells: about ±16 kPa (as in the example with well 2)

Solution

The oil line of well 3: $G_o = 7.04$ kPa/m, the same oil as in the new well.
At a depth of 2500 m it lies 100 kPa to the left of the oil line of the new well. This is 6 times the error of comparing wells. The oil of the block is a separate system.
The water-bearing points of well 3 differ from the water line of the new well by no more than 1.1 kPa. This is within the error of comparison, so the aquifer is probably common.
The FWL of the block is the crossing of its oil line with the common water line: 2510.9 m, which is 29.3 m higher than in the main block. The error is about ±5 m, because different wells are compared.

Answer: The oil behind the fault forms a separate system with an FWL at about 2510.9 m, which is about 29 m higher than in the main block, with a common aquifer.

What it means. The fault holds oil but apparently lets water through. This is how a capillary seal works (its pores are too small for the oil to enter): its entry pressure is greater than the pressure that the oil column creates. Another possibility is that the block was filled only up to its own spill point (the lowest point through which oil can leave). For hydrocarbons in place these are two different volumes of oil with their own contacts. The OWC from logs in well 3 should be at about 2509 m, if the rocks are the same. If the water-bearing points of well 3 had been shifted too, the aquifers would be separate, and the FWL of the block would have to be found from its own water-bearing measurements. The data in this example are assumed for illustration.

CalculatedExcess pressure over the common water line: the oil of the main block (the new well) and the oil of the block of well 3. The parallel oil lines are shifted by 0.1 MPa, so they cross the water line at different depths.

Tall accumulations and a change of composition with depth. In accumulations hundreds of metres tall, the composition of the fluid may change regularly with depth: heavy components concentrate at the bottom (compositional grading; Types of reservoir fluids). The density of the oil grows with depth, and the oil line bends slightly. A straight line through its upper part, extended downwards, is wrong by more the farther it is extended. In systems near the critical point (the state in which gas and liquid become identical), a gas condensate at the top can pass smoothly, with no clear boundary, into a volatile oil below. The “GOC” on the plot is then just a bend. The contrast of gradients is small (the table above), and the position of the transition is known only roughly.

An impossible gradient. A water gradient larger than hydrostatic in a uniform formation means upward flow, and a smaller one means downward flow. An oil line with the gradient of water means immobile oil or a wrong classification of points. Non-equilibrium systems (perched and tilted contacts, flow in the aquifer) are covered in the topic Perched and tilted contacts, trapped water, and pressure profiles in producing formations in the topic Pressure measurements in producing reservoirs: barriers and accuracy.

Terms and symbols

Free water level (FWL): The level where the pressures of oil (gas) and water are equal and the capillary pressure is zero. It is found from the crossing of the pressure lines of the phases.
OWC from logs: The boundary below which the rock is fully saturated with water. In a water-wet rock it lies above the FWL by $h_d = p_d/(\Delta\rho\,g)$ .
Recorded pressure: The pressure that the formation tester shows. In a water-wet rock it is the pressure of continuous oil above the OWC, and of water below it.
Excess pressure plot: The difference between the measured pressure and a reference line (usually the water line), against depth. It makes offsets of a few kilopascals visible.
Paleo-contact: The former position of a contact, marked by residual oil saturation below the present FWL. The log sees hydrocarbons there, and the pressure shows the water line.
Hydraulically isolated compartment: A part of an accumulation with no connection to its neighbours. On the pressure plot it has its own shifted line and its own contacts.
$\sigma_{z^*}$: The standard error of the depth of the crossing of two pressure lines.

Common mistakes and misconceptions

Wrong

“The crossing of the pressure lines is the OWC, and the log must show the same.”

In fact

The crossing is the FWL. The OWC from logs lies above it by a height that corresponds to the entry pressure of the rock: a fraction of a metre in a good sandstone, and metres in a tight rock.

Wrong

“A gradient can be found from three points, since a line passes through them.”

In fact

With one degree of freedom the scatter of the points cannot be estimated. The 95% interval of density from three points is several times wider than from five on the same interval. We need four or five valid points per line and as wide an interval as possible.

Wrong

“The gauge is accurate to hundredths of a kilopascal, so the FWL from the regional water line is known to centimetres.”

In fact

With lines from different wells, the error includes the absolute accuracy of the gauges and of the depth matching: tens of kilopascals and metres. Within one well these errors are common and cancel.

Wrong

“The log shows oil down to 2551 m, so the oil is mobile down to 2551 m.”

In fact

If the pressure in this interval falls on the water line, the oil in it is disconnected and immobile: this is residual oil below a paleo-contact. A sample or a test confirms this conclusion.

Wrong

“The pressure lines in two blocks coincide, so the blocks communicate and will be developed as one.”

In fact

Coincidence shows only that the pressures equalised over geological time. Connection over the years of development is checked by well tests and by watching the pressure.

In depth: advanced detail

Where the formula for the contact error comes from

A line fitted by least squares passes through the point $(\bar z, \bar p)$ and has the slope $G$ . The pressure on it at depth $z$ is $\bar p + G\,(z - \bar z)$ . The mean $\bar p$ has the variance $s^2/n$ , and the slope has the variance $s^2/\sum(z_i - \bar z)^2$ . When we measure from the mean depth, these two estimates are independent. So the variance of the pressure on the line is $\sigma^2(z) = s^2[1/n + (z - \bar z)^2/\sum(z_i - \bar z)^2]$ .

The crossing of two lines is the root of the equation $p_a(z) - p_b(z) = 0$ . If the pressures on the lines are shifted by small amounts $\delta p_a$ and $\delta p_b$ , the root shifts by $(\delta p_a - \delta p_b)/(G_b - G_a)$ : the pressure difference changes with depth at the rate $G_b - G_a$ . The errors of the two lines are independent, so the variances add up, and we get the formula of the main section. The errors of the slopes are included in it through the second term in the bracket, the shift of the level of the line at the depth of the crossing.

Systematic errors add up differently. A constant correction of the gauge $\delta$ , common to all the points of a survey, changes both lines by $\delta$ , and their difference does not change. A common shift of depths $\Delta z$ moves the whole picture down or up. The crossing shifts with it, but relative to the log of the same well, matched by the gamma ray log, it stays in place. If the lines are taken from different wells, each shifts in its own way. Then we add to the numerator the variances of $\delta$ of both wells and $(G\,\sigma_z)^2$ for the depths of each of them.

A GOC that the well did not penetrate: bubble point pressure

The gas–oil contact is found from the crossing of the gas and oil lines, but the gas cap has not been penetrated everywhere. In an equilibrium accumulation, the oil at the GOC is saturated with gas: its bubble point pressure (the pressure at which the first gas bubbles come out of the oil) equals the pressure at the contact. Suppose a sample of oil was taken, and the composition of the oil does not change with height. Then the GOC is the depth where the oil line reaches the bubble point pressure of the sample. If the pressure on the oil line at the crest of the structure (its highest point) is above the bubble point pressure, there is no gas cap.

GOC from the oil line and the bubble point pressure

z_{\text{GOC}} = \bar z_o + \frac{p_{\text{b}} - \bar p_o}{G_o},\qquad \sigma_{z_{\text{GOC}}} \approx \frac{\sigma_{p_{\text{b}}}}{G_o}

where:

$\bar z_o,\ \bar p_o$ mean depth and pressure of the points of the oil line, m, Pa
$p_{\text{b}}$ bubble point pressure of the oil from the sample, Pa
$G_o$ oil gradient, Pa/m
$\sigma_{p_{\text{b}}}$ error of the bubble point pressure (laboratory and sampling), Pa
$\sigma_{z_{\text{GOC}}}$ standard error of the GOC that we find (the error of the oil line itself is much smaller), m

Units: any consistent system

Holds when

The accumulation is in equilibrium, and the gas cap and the oil are connected.
The composition of the oil is constant with height. With compositional grading, the bubble point pressure of a sample taken below the GOC is smaller than that of the oil at the contact, and the GOC from this formula comes out above the true one.

The divisor is the oil gradient, not the difference of gradients. So each 70 kPa of error in the bubble point pressure gives about 10 m of error in the contact. The method answers reliably whether there is a gas cap, but it gives the place of the contact only roughly.

Worked exampleIs there a gas cap above the new well

Given (assumed data)

Oil line of the new well: $\bar z_o = 2501.06$ m, $\bar p_o = 27.67893$ MPa, $G_o = 7.065$ kPa/m
Oil sample from 2503.5 m: bubble point pressure 27.42 MPa, error ±0.5%
Crest of the structure: 2385 m

Solution

At the crest the oil line would give $27.67893 + 0.007065\cdot(2385 - 2501.06) = 26.859$ MPa, which is below the bubble point pressure. Single-phase oil cannot exist at such a pressure, so there is a gas cap higher on the structure.
The depth where the oil line reaches 27.42 MPa: $2501.06 + (27.42 - 27.67893)/0.007065 = 2464$ m.
The error: $0.005\cdot 27.42 = 0.137$ MPa, and $137/7.065 = 19$ m.

Answer: There is a gas cap. The GOC is about 2464 ± 19 m, in agreement with the crossing of the gas and oil lines (about 2465 m) in the topic on reservoir pressure.

What it means. The crossing of the pressure lines finds the GOC an order of magnitude (about ten times) more accurately, but it needs measurements in the gas cap. The bubble point pressure confirms that a cap exists, and it gives an independent check. If the composition of the oil changes with height, we first check at what depth the sample was taken, and how much the bubble point pressure changes through the section. The data in this example are assumed for illustration.

Oil-wet rock: the mirror picture

In a strongly oil-wet rock the wetting phase is oil. The capillary pressure, defined as the pressure of the oil minus the pressure of the water, is negative. The transition zone lies below the FWL: the oil is “drawn” down into the water-bearing part, against buoyancy (the upward push on the lighter fluid). The argument about the probe as a large “pore” gives the mirror answer. In the transition zone this large pore is filled by the non-wetting phase, which is water, and the tool shows the water pressure. So in an oil-wet rock the water line extends up to the FWL, and the log sees oil also below the crossing of the lines. Unlike the residual oil below a paleo-contact, this oil is mobile.

In practice, strong oil-wetness in the transition zone itself is rare. The oil-bearing part high above the contact is often oil-wet or of mixed wettability (some of the rock oil-wet and some water-wet; Wettability and interfacial phenomena). At the contact, where there is a lot of water, the rock usually stays water-wet. So when the log sees hydrocarbons below the crossing of the lines, the first hypothesis is residual oil, and the second is oil-wetness. A sample and a test of the interval tell them apart.

Connections to other topics

Pressure measurement with the wireline formation tester — pretests, their classes, the accuracy of measurements and the gradient from a set of points.
Reservoir pressure and temperature — fluid gradients, contacts from two points per zone, the pressure at the crest of an accumulation.
Capillary pressure and the transition zone — entry pressure, the transition zone and why the OWC is above the FWL.
Supercharging and permeability from formation-tester pretests — supercharging of tight layers and how it shifts lines and contacts.
Forced-gradient technique and multi-well analysis — lines with a known gradient, and joint processing of several wells.
Perched and tilted contacts, trapped water — perched and tilted contacts, flow in the aquifer.
Gross rock volume from maps — how the position of a contact turns into the rock volume of an accumulation.
Faults — sealing faults and hydraulically isolated compartments.
Types of reservoir fluids — change of fluid composition with depth in tall accumulations.

Check yourself

1
A pretest 1.5 m below the OWC from logs, but 0.8 m above the FWL found from the crossing of lines. On which line should its point fall in a water-wet rock?
1. A.The point is always invalid
2. B.On the oil line
3. C.On the water line
4. D.Midway between the lines
Show the answer
Correct answer: C. On the water line
Below the OWC the rock is fully saturated with water: the capillary pressure there is smaller than the entry pressure, and oil has not entered it. The tool shows the water pressure, although the oil line, extended downwards, is still to the right of the water line at this depth.
2
The oil line: $p = 24.316 + 0.00695\,(z - 2300)$ MPa. The water line from another part of the same well: $p = 24.874 + 0.01030\,(z - 2360)$ MPa. Find the FWL.
Show the answer
At 2360 m the oil is at $24.316 + 0.00695\cdot 60 = 24.733$ MPa, and the water is at 24.874 MPa: the water is higher by 0.141 MPa. The difference of gradients is $10.30 - 6.95 = 3.35$ kPa/m, so the lines meet $141/3.35 = 42.1$ m higher. The FWL is at about 2317.9 m.
3
At the depth of the crossing, the standard errors of the oil and water lines are 1.2 and 1.6 kPa, and the difference of the gradients is 2.5 kPa/m. What is the standard error of the FWL, and what will reduce it most, if the points of the water line lie 30 m below the crossing?
Show the answer
$\sqrt{1.2^2 + 1.6^2}/2.5 = 2.0/2.5 = 0.8$ m. The error of the water line at the depth of the crossing is large because of the extrapolation by 30 m. So additional pretests in the water-bearing part closer to the contact will help most, not new points far from it.
4
From the resistivity log, oil is seen down to 2551 m, but the pretests from 2541 to 2556 m lie on the water line. The oil and water lines cross at 2540.5 m. What is the most likely explanation?
1. A.Capillary forces lowered the OWC below the FWL in a water-wet rock
2. B.The oil in the interval 2540–2551 m is disconnected and immobile: residual oil below a paleo-contact
3. C.The FWL is actually at 2551 m, and the pretests are wrong
4. D.Supercharging of water-bearing layers
Show the answer
Correct answer: B. The oil in the interval 2540–2551 m is disconnected and immobile: residual oil below a paleo-contact
In a water-wet rock the OWC can only be above the FWL. Oil that the log sees below the crossing of the lines, with the pressure on the water line, is disconnected droplets left after the contact rose or the accumulation was rearranged. This is checked with a formation tester sample or a test of the interval. If the rock is oil-wet, the oil could also be mobile.
5
Why is the GOC of light oil and gas from the crossing of lines usually found more accurately than the FWL of heavy oil, with the same accuracy of measurements?
Show the answer
The error of a contact is inversely proportional to the difference of the gradients of the phases. For oil of 0.72 and gas of 0.20 g/cm³ the difference is about 5.1 kPa/m. For water of 1.07 and oil of 0.90 g/cm³ it is about 1.7 kPa/m. Each kilopascal of error shifts the GOC by about 0.2 m, and the FWL of heavy oil by 0.6 m.
6
The oil line of a well in a neighbouring block is parallel to the line of the main block, but 60 kPa to the right. The error of comparing the wells is about 15 kPa. According to the data of both wells, the aquifer is common. Where is the FWL of this block relative to the main one, if the difference of the water and oil gradients is 3.4 kPa/m?
1. A.It cannot be found in principle
2. B.Deeper by about 18 m
3. C.At the same depth
4. D.Higher by about 18 m
Show the answer
Correct answer: B. Deeper by about 18 m
A shift of 60 kPa is four times the error. The block is a separate system. The oil line to the right (a higher pressure) crosses the common water line lower, by $60/3.4 \approx 18$ m. The oil column in this block is taller, for example because its spill point is deeper.

Updated: 16 October 2026